lesson
An identity holds throughout its domain
1 · Learn
An equation may hold only for selected values. An identity holds for every permitted value of its variables. Expanding 3(x + 2) gives 3x + 6, so those expressions are identical for real x.
Testing one substitution can disprove a false identity but cannot prove a general identity. Use algebraic transformations with stated restrictions to establish equivalence.
2 · Worked example
3(x + 2) = 18 has solution x = 4. By contrast, 3(x + 2) = 3x + 6 holds for every real x.
3 · Your turn
Is 2(x + 3) = 2x + 6 an identity? Explain.
Check your answer
Yes; distributing 2 gives 2x + 6 for every real x.
4 · Apply your learning
Find a counterexample to the false claim (x + 2)² = x² + 4, then expand correctly.
Adult guidance and safety
Check prerequisites, units and reasoning. Provide accurate diagrams and varied practice. Extension content needs suitable prior understanding; no GCSE board, tier or examination readiness is implied by this proposed sequence.
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Check a calculation · Year 5 onward
Use this only when your teacher or adult allows a calculator for the task. Show your thinking first. It does not replace mental or written arithmetic and does not mark your lesson answer.
Powers, roots and trigonometry · Year 7 onward
Use only when the task allows a calculator. Choose one operation at a time and record your working. Trigonometry here always uses degrees (DEG), not radians. Inverse sine, cosine and tangent return an angle, not a reciprocal. Keep exact fractions and π in answers when requested.
Approximate numerical results use up to 12 significant digits. Inverse sine returns −90° to 90°, inverse cosine 0° to 180°, and inverse tangent −90° to 90°; these are principal values, not every possible solution of a trigonometric equation.
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